Dynamics of the family of c-iterative methods

In this paper, the dynamics of the family of c-iterative methods for solving nonlinear equations are studied on quadratic polynomials. A singular parameter space is presented to show the complexity of the family. The analysis of the parameter space allows us to find elements of the family that have bad convergence properties and also other ones with very stable behaviour. These schemes correspond to values of c in different small regions of the parameter space.Supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02. The first and fourth authors were also partially supported by P11B2011-30 (Universitat Jaume I), the second and third authors were also partially supported by Vicerrectorado de Investigacion, Universitat Politecnica de Valencia PAID-06-2010-2285.Campos, B.; Cordero Barbero, A.; Torregrosa Sánchez, JR.; Vindel, P. (2015). Dynamics of the family of c-iterative methods. International Journal of Computer Mathematics. 92(9):1815-1825. https://doi.org/10.1080/00207160.2014.893608S1815182592

Orbits of period two in the family of a multipoint variant of Chebyshev-Halley family

[EN] The study of the dynamical behaviour of the operators defined by iterative methods help us to know more deeply the regions where these methods have a good performance. In this paper, we follow the dynamical study of a multipoint variant of the known Chebyshev-Halley's family, showing the existence of attractive periodic orbits of period 2 for some values of the parameter.This research was partially supported by Ministerio de Econom´ı a y Competitividad MTM2014-52016-C02-2-PCampos, B.; Cordero Barbero, A.; Torregrosa Sánchez, JR.; Vindel Cañas, P. (2016). Orbits of period two in the family of a multipoint variant of Chebyshev-Halley family. Numerical Algorithms. 73(1):141-156. https://doi.org/10.1007/s11075-015-0089-0141156731Blanchard, P.: Complex analytic dynamics on the Riemann sphere. Bull. AMS 11(1), 85–141 (1984)Beardon, A.F.: Iteration of Rational Functions, Graduate Texts in Mathematics. Springer-Verlag, New York (1991)Behl, R., Kanwar, V.: Variants of Chebyshev’s method with optimal order of convergence. Tamsui Oxf. J. Inf. Math. Sci. 29(1), 39–53 (2013)Campos, B., Cordero, A., Magreñan, A., Torregrosa, J.R., Vindel, P.: Study of a bi-parametric family of iterative methods. Abstr. Appl. Anal. 2014. Art. ID 141643, 12 ppCampos, B., Cordero, A., Torregrosa, J.R., Vindel, P.: Bifurcations in the dynamics of a variant of Chebyshev method. In: Proceedings of the 15th International Conference on Computational and Mathematical Methods in Science and Engineering CMMSE 2015, ISBN 978-84-617-2230-3, pp. 291–299 (2015)Chicharro, F., Cordero, A., Torregrosa, J.R.: Drawing dynamical and parameter planes of iterative families and methods. The Scientific World Journal Volume 2013 Article ID 780153Varona, J.L.: Graphic and numerical comparison between iterative methods. Math. Intelligencer 24, 37–46 (2002)Amat, S., Busquier, S., Plaza, S.: Review of some iterative root-finding methods from a dynamical point of view. Sci. Ser. A: Math. Sci. 10, 3–35 (2004)Cordero, A., García-Maimó, J., Torregrosa, J.R., Vassileva, M.P., Vindel, P.: Chaos in King’s iterative family. Appl. Math. Lett. 26, 842–848 (2013)Cordero, A., Torregrosa, J.P., Vindel, P.: Dynamics of a family of Chebyshev-Halley type method. Appl. Math. Comput. 219, 8568–8583 (2013)Gutiérrez, J.M., Hernández, M.A., Romero, N.: Dynamics of a new family of iterative processes for quadratic polynomials. J. Comput. Appl. Math. 233, 2688–2695 (2010)Neta, B., Chun, C., Scott, M.: Basins of attraction for optimal eighth order methods to find simple roots of nonlinear equation. App. Math. Comput. 227, 567–592 (2014)Scott, M., Neta, B., Chun, C.: Basin attractors for various methods. Appl. Math. Comput. 218, 2584–2599 (2011)Blanchard, P.: The dynamics of Newton’s method. Proc. Symp. Appl. Math. 49, 139–154 (1994)Cordero, A., Torregrosa, J.R.: Variants of Newton’s method using fifth-order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007

Behavior of fixed and critical points of the (alpha,c)-family of iterative methods

In this paper we study the dynamical behavior of the -family of iterative methods for solving nonlinear equations, when we apply the fixed point operator associated to this family on quadratic polynomials. This is a family of third-order iterative root-finding methods depending on two parameters; so, as we show throughout this paper, its dynamics is really interesting, but complicated. In fact, we have found in the real -plane a line in which the corresponding elements of the family have a lower number of free critical points. As this number is directly related with the quantity of basins of attraction, it is probable to find more stable behavior between the elements of the family in this region.Supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02. The first and fourth authors were also partially supported by P11B2011-30 (Universitat Jaume I), the second and third authors were also partially supported by Vicerrectorado de Investigacion, Universitat Politecnica de Valencia SP20120474.Campos, B.; Cordero Barbero, A.; Torregrosa Sánchez, JR.; P. Vindel (2015). Behavior of fixed and critical points of the (alpha,c)-family of iterative methods. Journal of Mathematical Chemistry. 53(3):807-827. https://doi.org/10.1007/s10910-014-0465-3S807827533A.F. Beardon, Iteration of Rational Functions, Graduate Texts in Mathematics, vol. 132 (Springer, New York, 1991)B. Campos, A. Cordero, A. Magreñan, J.R. Torregrosa, P. Vindel, Study of a bi-parametric family of iterative methods, Abstra. Appl. Anal. (2014). doi: 10.1155/2014/141643B. Campos, A. Cordero, A. Magreñan, J.R. Torregrosa, P. Vindel, Bifurcations of the roots of a 6-degree symmetric polynomial coming from the fixed point operator of a class of iterative methods. in Proceedings of the 14th International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE, ed. by J. Vigo-Aguiar (2014), pp. 253–264B. Campos, A. Cordero, J.R. Torregrosa, P. Vindel, Dynamics of the family of c-iterative methods. Int. J. Compt. Math. (2014). doi: 10.1080/00207160.2014.893608F. Chicharro, A. Cordero, J.R. Torregrosa, Drawing dynamical and parameter planes of iterative families and methods. Sci. World J. (2013). doi: 10.1155/2013/780153A. Cordero, J. García-Maimó, J.R. Torregrosa, M.P. Vassileva, P. Vindel, Chaos in King’s iterative family. Appl. Math. Lett. 26, 842–848 (2013)A. Cordero, J.R. Torregrosa, P. Vindel, Dynamics of a family of Chebyshev–Halley type method. Appl. Math. Comput. 219, 8568–8583 (2013)C.G. Jesudason, I. Numerical nonlinear analysis: differential methods and optimization applied to chemical reaction rate determination. J. Math. Chem. 49(7), 1384–1415 (2011)P.G. Logrado, J.D.M. Vianna, Partitioning technique procedure revisited: formalism and first applications to atomic problems. J. Math. Chem. 22, 107–116 (1997)M. Mahalakshmi, G. Hariharan, K. Kannan, The wavelet methods to linear and nonlinear reaction-diffusion model arising in mathematical chemistry. J. Math. Chem. 51(9), 2361–2385 (2013)K. Maleknejad, M. Alizadeh, An efficient numerical sheme for solving Hammerstein integral equation arisen in chemical phenomenon. Proc. Comput. Sci. 3, 361–364 (2011)J. Milnor, Dynamics in One Complex Variable (Princeton University Press, New Jersey, 2006

Real qualitative behavior of a fourth-order family of iterative methods by using the convergence plane

The real dynamics of a family of fourth-order iterative methods is studied when it is applied on quadratic polynomials. A Scaling Theorem is obtained and the conjugacy classes are analyzed. The convergence plane is used to obtain the same kind of information as from the parameter space, and even more, in complex dynamics. (C) 2014 IMACS. Published by Elsevier B.V. All rights reserved.This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-{01,02}.Magreñán Ruiz, ÁA.; Cordero Barbero, A.; Gutiérrez Jiménez, JM.; Torregrosa Sánchez, JR. (2014). Real qualitative behavior of a fourth-order family of iterative methods by using the convergence plane. Mathematics and Computers in Simulation. 105:49-61. https://doi.org/10.1016/j.matcom.2014.04.006S496110

Convergence, efficiency and dynamics of new fourth and sixth order families of iterative methods for nonlinear system

[EN] In this work we present a new family of iterative methods for solving nonlinear systems that are optimal in the sense of Kung and Traub’s conjecture for the unidimensional case. We generalize this family by performing a new step in the iterative method, getting a new family with order of convergence six. We study the efficiency of these families for the multidimensional case by introducing a new term in the computational cost defined by Grau-Sánchez et al. A comparison with already known methods is done by studying the dynamics of these methods in an example system.This research has been supported by Ministerio de Ciencia e Innovacion MTM2011-28636-C02-02 and by Vicerrectorado de Investigacion Universitat Politecnica de Valencia PAID-SP-2012-0498 and PAID-SP-2012-0474.Hueso Pagoaga, JL.; Martínez Molada, E.; Teruel, C. (2015). Convergence, efficiency and dynamics of new fourth and sixth order families of iterative methods for nonlinear system. Journal of Computational and Applied Mathematics. 275:420-412. https://doi.org/10.1016/j.cam.2014.06.010S42041227

Chaos and convergence of a family generalizing Homeier's method with damping parameters

[EN] In this paper, a family of parametric iterative methods for solving nonlinear equations, including Homeier's scheme, is presented. Its local convergence is obtained and the dynamical behavior on quadratic polynomials of the resulting family is studied in order to choose those values of the parameter that ensure stable behavior. To get this aim, the analysis of fixed and critical points and the associated parameter plane show the dynamical richness of the family and allow us to find members of this class with good numerical properties and also other ones with pathological conduct. To check the stable behavior of the good selected ones, the discretized planar 1D-Bratu problem is solved. Some of those chosen members of the family achieve good results when Homeier's scheme fails.This research was supported by Ministerio de Economia y Competitividad MTM2014-52016-C02-2-P.Cordero Barbero, A.; Franques, A.; Torregrosa Sánchez, JR. (2016). Chaos and convergence of a family generalizing Homeier's method with damping parameters. Nonlinear Dynamics. 85(3):1939-1954. https://doi.org/10.1007/s11071-016-2807-0S19391954853Amat, S., Busquier, S., Bermúdez, C., Magreñán, Á.A.: On the election of the damped parameter of a two-step relaxed Newton-type method. Nonlinear Dyn. doi: 10.1007/s11071-015-2179-xAmat, S., Busquier, S., Bermúdez, C., Plaza, S.: On two families of high order Newton type methods. Appl. Math. Lett. 25, 2209–2217 (2012)Amat, S., Busquier, S., Plaza, S.: Review of some iterative root-finding methods from a dynamical point of view. Sci. Ser. A Math. Sci. 10, 3–35 (2004)Babajee, D.K.R., Cordero, A., Torregrosa, J.R.: Study of iterative methods through the Cayley Quadratic Test. J. Comput. Appl. Math. 291, 358–369 (2016)Babajee, D.K.R., Thukral, R.: On a 4-point sixteenth-order king family of iterative methods for solving nonlinear equations. Int. J. Math. Math. Sci. 2012, ID 979245, 13 (2012)Blanchard, P.: Complex analytic dynamics on the Riemann sphere. Bull. AMS 11(1), 85–141 (1984)Blanchard, P.: The dynamics of Newton’s method. Proc. Symp. Appl. Math. 49, 139–154 (1994)Boyd, J.P.: One-point pseudospectral collocation for the one-dimensional Bratu equation. Appl. Math. Comput. 217, 5553–5565 (2011)Bratu, G.: Sur les equation integrals non-lineaires. Bull. Math. Soc. Fr. 42, 113–142 (1914)Chicharro, F., Cordero, A., Torregrosa, J.R.: Drawing dynamical and parameter planes of iterative families and methods. Sci. World J. 2013, Article ID 780153 (2013)Chun, C., Lee, M.Y.: A new optimal eighth-order family of iterative methods for the solution of nonlinear equations. Appl. Math. Comput. 223, 506–519 (2013)Cordero, A., García-Maimó, J., Torregrosa, J.R., Vassileva, M.P., Vindel, P.: Chaos in King’s iterative family. Appl. Math. Lett. 26, 842–848 (2013)Cordero, A., Torregrosa, J.R.: Variants of Newton’s method using fifth-order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007)Cordero, A., Torregrosa, J.R., Vindel, P.: Dynamics of a family of Chebyshev–Halley type method. Appl. Math. Comput. 219, 8568–8583 (2013)Fatou, P.: Sur les équations fonctionnelles. Bull. Soc. Math. Fr. 47, 161–271 (1919); 48, 33–94; 208–314 (1920)Gelfand, I.M.: Some problems in the theory of quasi-linear equations. Transl. Am. Math. Soc. Ser. 2, 295–381 (1963)Gutiérrez, J.M., Hernández, M.A., Romero, N.: Dynamics of a new family of iterative processes for quadratic polynomials. J. Comput. Appl. Math. 233, 2688–2695 (2010)Homeier, H.H.H.: On Newton-type methods with cubic convergence. J. Comput. Appl. Math. 176, 425–432 (2005)Jacobsen, J., Schmitt, K.: The Liouville–Bratu–Gelfand problem for radial operators. J. Differ. Equ. 184, 283–298 (2002)Jalilian, R.: Non-polynomial spline method for solving Bratu’s problem. Comput. Phys. Commun. 181, 1868–1872 (2010)Julia, G.: Mémoire sur l’iteration des fonctions rationnelles. J. Math. Pure Appl. 8, 47–245 (1918)Magreñán, Á.A.: Different anomalies in a Jarratt family of iterative root-finding methods. Appl. Math. Comput. 233, 29–38 (2014)Mohsen, A.: A simple solution of the Bratu problem. Comput. Math. Appl. 67, 26–33 (2014)Neta, B., Chun, C., Scott, M.: Basins of attraction for optimal eighth order methods to find simple roots of nonlinear equation. Appl. Math. Comput. 227, 567–592 (2014)Ostrowski, A.M.: Solution of Equations and Systems of Equations. Academic Press, New York (1960)Petković, M., Neta, B., Petković, L.D., Džunić, J.: Multipoint Methods for Solving Nonlinear Equations. Academic Press, Amsterdam (2013)Scott, M., Neta, B., Chun, C.: Basin attractors for various methods. Appl. Math. Comput. 218, 2584–2599 (2011)Sharma, J.R.: Improved Chebyshev–Halley method with sixth and eighth order of convergence. Appl. Math. Comput. 256, 119–124 (2015)Varona, J.L.: Graphic and numerical comparison between iterative methods. Math. Intell. 24, 37–46 (2002)Wan, Y.Q., Guo, Q., Pan, N.: Thermo-electro-hydrodynamic model for electrospinning process. Int. J. Nonlinear Sci. Numer. Simul. 5, 5–8 (2004

Widening basins of attraction of optimal iterative methods

[EN] In this work, we analyze the dynamical behavior on quadratic polynomials of a class of derivative-free optimal parametric iterative methods, designed by Khattri and Steihaug. By using their parameter as an accelerator, we develop different methods with memory of orders three, six and twelve, without adding new functional evaluations. Then a dynamical approach is made, comparing each of the proposed methods with the original ones without memory, with the following empiric conclusion: Basins of attraction of iterative schemes with memory are wider and the behavior is more stable. This has been numerically checked by estimating the solution of a practical problem, as the friction factor of a pipe and also of other nonlinear academic problems.This research was supported by Islamic Azad University, Hamedan Branch, Ministerio de Economia y Competitividad MTM2014-52016-C02-2-P and Generalitat Valenciana PROMETEO/2016/089.Bakhtiari, P.; Cordero Barbero, A.; Lotfi, T.; Mahdiani, K.; Torregrosa Sánchez, JR. (2017). Widening basins of attraction of optimal iterative methods. Nonlinear Dynamics. 87(2):913-938. https://doi.org/10.1007/s11071-016-3089-2S913938872Amat, S., Busquier, S., Bermúdez, C., Plaza, S.: On two families of high order Newton type methods. Appl. Math. Lett. 25, 2209–2217 (2012)Amat, S., Busquier, S., Bermúdez, C., Magreñán, Á.A.: On the election of the damped parameter of a two-step relaxed Newton-type method. Nonlinear Dyn. 84(1), 9–18 (2016)Chun, C., Neta, B.: An analysis of a family of Maheshwari-based optimal eighth order methods. Appl. Math. Comput. 253, 294–307 (2015)Babajee, D.K.R., Cordero, A., Soleymani, F., Torregrosa, J.R.: On improved three-step schemes with high efficiency index and their dynamics. Numer. Algorithms 65(1), 153–169 (2014)Argyros, I.K., Magreñán, Á.A.: On the convergence of an optimal fourth-order family of methods and its dynamics. Appl. Math. Comput. 252, 336–346 (2015)Petković, M., Neta, B., Petković, L., Džunić, J.: Multipoint Methods for Solving Nonlinear Equations. Academic Press, London (2013)Ostrowski, A.M.: Solution of Equations and System of Equations. Prentice-Hall, Englewood Cliffs, NJ (1964)Kung, H.T., Traub, J.F.: Optimal order of one-point and multipoint iteration. J. ACM 21, 643–651 (1974)Khattri, S.K., Steihaug, T.: Algorithm for forming derivative-free optimal methods. Numer. Algorithms 65(4), 809–824 (2014)Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice Hall, New York (1964)Cordero, A., Soleymani, F., Torregrosa, J.R., Shateyi, S.: Basins of Attraction for Various Steffensen-Type Methods. J. Appl. Math. 2014, 1–17 (2014)Devaney, R.L.: The Mandelbrot Set, the Farey Tree and the Fibonacci sequence. Am. Math. Mon. 106(4), 289–302 (1999)McMullen, C.: Families of rational maps and iterative root-finding algorithms. Ann. Math. 125(3), 467–493 (1987)Chicharro, F., Cordero, A., Gutiérrez, J.M., Torregrosa, J.R.: Complex dynamics of derivative-free methods for nonlinear equations. Appl. Math. Comput. 219, 70237035 (2013)Magreñán, Á.A.: Different anomalies in a Jarratt family of iterative root-finding methods. Appl. Math. Comput. 233, 29–38 (2014)Neta, B., Chun, C., Scott, M.: Basins of attraction for optimal eighth order methods to find simple roots of nonlinear equations. Appl. Math. Comput. 227, 567–592 (2014)Lotfi, T., Magreñán, Á.A., Mahdiani, K., Rainer, J.J.: A variant of Steffensen–King’s type family with accelerated sixth-order convergence and high efficiency index: dynamic study and approach. Appl. Math. Comput. 252, 347–353 (2015)Chicharro, F.I., Cordero, A., Torregrosa, J.R.: Drawing dynamical and parameters planes of iterative families and methods. Sci. World J. 2013, 1–11 (2013)Cordero, A., Lotfi, T., Torregrosa, J.R., Assari, P., Mahdiani, K.: Some new bi-accelerator two-point methods for solving nonlinear equations. Comput. Appl. Math. 35(1), 251–267 (2016)Cordero, A., Lotfi, T., Bakhtiari, P., Torregrosa, J.R.: An efficient two-parametric family with memory for nonlinear equations. Numer. Algorithms 68(2), 323–335 (2015)Lotfi, T., Mahdiani, K., Bakhtiari, P., Soleymani, F.: Constructing two-step iterative methods with and without memory. Comput. Math. Math. Phys. 55(2), 183–193 (2015)Cordero, A., Maimó, J.G., Torregrosa, J.R., Vassileva, M.P.: Solving nonlinear problems by Ostrowski–Chun type parametric families. J. Math. Chem. 53, 430–449 (2015)Abad, M., Cordero, A., Torregrosa, J.R.: A family of seventh-order schemes for solving nonlinear systems. Bull. Math. Soc. Sci. Math. Roum. Tome 57(105), 133–145 (2014)Weerakoon, S., Fernando, T.G.I.: A variant of Newton’s method with accelerated third-order convergence. Appl. Math. Lett. 13, 87–93 (2000)White, F.: Fluid Mechanics. McGraw-Hill, Boston (2003)Zheng, Q., Li, J., Huang, F.: An optimal Steffensen-type family for solving nonlinear equations. Appl. Math. Comput. 217, 9592–9597 (2011)Soleymani, F., Babajee, D.K.R., Shateyi, S., Motsa, S.S.: Construction of optimal derivative-free techniques without memory. J. Appl. Math. (2012). doi: 10.1155/2012/49702

Local convergence of a family of iterative methods for Hammerstein equations

[EN] In this paper we give a local convergence result for a uniparametric family of iterative methods for nonlinear equations in Banach spaces. We assume boundedness conditions involving only the first Fr,chet derivative, instead of using boundedness conditions for high order derivatives as it is usual in studies of semilocal convergence, which is a drawback for solving some practical problems. The existence and uniqueness theorem that establishes the convergence balls of these methods is obtained. We apply this theory to different examples, including a nonlinear Hammerstein equation that have many applications in chemistry and appears in problems of electro-magnetic fluid dynamics or in the kinetic theory of gases. With these examples we illustrate the advantages of these results. The global convergence of the method is addressed by analysing the behaviour of the methods on complex polynomials of second degree.This research was supported by Ministerio de Ciencia y Tecnologia MTM2014-52016-C2-02.This research was supported by Ministerio de Ciencia y Tecnología MTM2014-52016-C2-02.Martínez Molada, E.; Singh, S.; Hueso Pagoaga, JL.; Gupta, D. (2016). Local convergence of a family of iterative methods for Hammerstein equations. Journal of Mathematical Chemistry. 54(7):1370-1386. https://doi.org/10.1007/s10910-016-0602-2S13701386547I.K. Argyros, S. Hilout, M.A. Tabatabai, Mathematical Modelling with Applications in Biosciences and Engineering (Nova Publishers, New York, 2011)J.F. Traub, Iterative Methods for the Solution of Equations (Prentice-Hall, Englewood Cliffs, New Jersey, 1964)A.M. Ostrowski, Solutions of Equations in Euclidean and Banach Spaces (Academic Press, New York, 1973)I.K. Argyros, J.A. Ezquerro, J.M. Gutiárrez, M.A. Hernández, S. Hilout, On the semilocal convergence of efficient ChebyshevSecant-type methods. J. Comput. Appl. Math. 235, 3195–3206 (2011)José L. Hueso, E. Martínez, Semilocal convergence of a family of iterative methods in Banach spaces. Numer. Algorithms 67, 365–384 (2014)X. Wang, C. Gu, J. Kou, Semilocal convergence of a multipoint fourth-order super-Halley method in Banach spaces. Numer. Algorithms 54, 497–516 (2011)J. Kou, Y. Li, X. Wang, A variant of super Halley method with accelerated fourth-order convergence. Appl. Math. Comput. 186, 535–539 (2007)L. Zheng, C. Gu, Recurrence relations for semilocal convergence of a fifth-order method in Banach spaces. Numer. Algorithms 59, 623–638 (2012)S. Amat, M.A. Hernández, N. Romero, A modified Chebyshevs iterative method with at least sixth order of convergence. Appl. Math. Comput. 206, 164–174 (2008)X. Wang, J. Kou, C. Gu, Semilocal convergence of a sixth-order Jarratt method in Banach spaces. Numer. Algorithms 57, 441–456 (2011)A. Cordero, J.A. Ezquerro, M.A. Hernández-Verón, J.R. Torregrosa, On the local convergence of a fifth-order iterative method in Banach spaces. Appl. Math. Comput. 251, 396–403 (2015)I.K. Argyros, S. Hilout, On the local convergence of fast two-step Newton-like methods for solving nonlinear equations. J. Comput. Appl. Math. 245, 1–9 (2013)S. Weerakoon, T.G.I. Fernando, A variant of Newton’s method with accelerated third-order convergence. Appl. Math. Lett. 13(8), 87–93 (2000)X. Feng, Y. He, High order oterative methods without derivatives for solving nonlinear equations. Appl. Math. Comput. 186, 1617–1623 (2007)X. Wang, J. Kou, Y. Li, Modified Jarratt method with sixth-order convergence. Appl. Math. Lett. 22, 1798–1802 (2009)A.D. Polyanin, A.V. Manzhirov, Handbook of Integral Equations (CRC Press, Boca Raton, 1998)S. Plaza, N. Romero, Attracting cycles for the relaxed Newton’s method. J. Comput. Appl. Math. 235(10), 3238–3244 (2011)A. Cordero, J.R. Torregrosa, P. Vindel, Study of the dynamics of third-order iterative methods on quadratic polynomials. Int. J. Comput. Math. 89(13–14), 1826–1836 (2012)Gerardo Honorato, Sergio Plaza, Natalia Romero, Dynamics of a higher-order family of iterative methods. J. Complex. 27(2), 221–229 (2011)J.M. Gutirrez, M.A. Hernández, N. Romero, Dynamics of a new family of iterative processes for quadratic polynomials. J. Comput. Appl. Math. 233(10), 2688–2695 (2010)I.K. Argyros, A.A. Magreñan, A study on the local convergence and dynamics of Chebyshev-Halley-type methods free from second derivative. Numer. Algorithms. doi: 10.1007/s11075-015-9981-xI.K. Argyros, S. George, Local convergence of modified Halley-like methods with less computation of inversion (Novi Sad J. Math, Draft version, 2015

CMMSE2017: On two classes of fourth- and seventh-order vectorial methods with stable behavior

[EN] A family of fourth-order iterative methods without memory, for solving nonlinear systems, and its seventh-order extension, are analyzed. By using complex dynamics tools, their stability and reliability are studied by means of the properties of the rational function obtained when they are applied on quadratic polynomials. The stability of their fixed points, in terms of the value of the parameter, its critical points and their associated parameter planes, etc. give us important information about which members of the family have good properties of stability and whether in any of them appear chaos in the iterative process. The conclusions obtained in this dynamical analysis are used in the numerical section, where an academical problem and also the chemical problem of predicting the diffusion and reaction in a porous catalyst pellet are solved.This research was partially supported by Ministerio de Economia y Competitividad MTM2014-52016-C02-2-P and Generalitat Valenciana PROMETEO/2016/089.Cordero Barbero, A.; Guasp, L.; Torregrosa Sánchez, JR. (2018). CMMSE2017: On two classes of fourth- and seventh-order vectorial methods with stable behavior. Journal of Mathematical Chemistry. 56(7):1902-1923. https://doi.org/10.1007/s10910-017-0814-0S19021923567S. Amat, S. Busquier, Advances in Iterative Methods for Nonlinear Equations (Springer, Berlin, 2016)S. Amat, S. Busquier, S. Plaza, Review of some iterative root-finding methods from a dynamical point of view. Sci. Ser. A Math. Sci. 10, 3–35 (2004)S. Amat, S. Busquier, S. Plaza, A construction of attracting periodic orbits for some classical third-order iterative methods. Comput. Appl. Math. 189, 22–33 (2006)I.K. Argyros, Á.A. Magreñn, On the convergence of an optimal fourth-order family of methods and its dynamics. Appl. Math. Comput. 252, 336–346 (2015)D.K.R. Babajee, A. Cordero, J.R. Torregrosa, Study of multipoint iterative methods through the Cayley quadratic test. Comput. Appl. Math. 291, 358–369 (2016). doi: 10.1016/J.CAM.2014.09.020P. Blanchard, The dynamics of Newton’s method. Proc. Symp. Appl. Math. 49, 139–154 (1994)F.I. Chicharro, A. Cordero, J.R. Torregrosa, Drawing dynamical and parameters planes of iterative families and methods. Sci. World J. 2013, Article ID 780153 (2013)C. Chun, M.Y. Lee, B. Neta, J. Džunić, On optimal fourth-order iterative methods free from second derivative and their dynamics. Appl. Math. Comput. 218, 6427–6438 (2012)A. Cordero, E. Gómez, J.R. Torregrosa, Efficient high-order iterative methods for solving nonlinear systems and their application on heat conduction problems. Complexity 2017, Article ID 6457532 (2017)A. Cordero, J.R. Torregrosa, Variants of Newton’s method using fifth-order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007)R.L. Devaney, An Introduction to Chaotic Dynamical Systems (Addison-Wesley Publishing Company, Reading, 1989)P.G. Logrado, J.D.M. Vianna, Partitioning technique procedure revisited: formalism and first application to atomic problems. Math. Chem. 22, 107–116 (1997)C.G. Jesudason, I. Numerical nonlinear analysis: differential methods and optimization applied to chemical reaction rate determination. Math. Chem. 49, 1384–1415 (2011)Á.A. Magreñán, Different anomalies in a Jarratt family of iterative root-finding methods. Appl. Math. Comput. 233, 29–38 (2014)M. Mahalakshmi, G. Hariharan, K. Kannan, The wavelet methods to linear and nonlinear reaction-diffusion model arising in mathematical chemistry. Math. Chem. 51(9), 2361–2385 (2013)K. Maleknejad, M. Alizadeh, An efficient numerical scheme for solving Hammerstein integral equation arisen in chemical phenomenon. Proc. Comput. Sci. 3, 361–364 (2011)B. Neta, C. Chun, M. Scott, Basins of attraction for optimal eighth-order methods to find simple roots of nonlinear equations. Appl. Math. Comput. 227, 567–592 (2014)M.S. Petković, B. Neta, L.D. Petković, J. Džunić, Multipoint Methods for Solving Nonlinear Equations (Elsevier, Amsterdam, 2013)R.C. Rach, J.S. Duan, A.M. Wazwaz, Solving coupled Lane–Emden boundary value problems in catalytic diffusion reactions by the Adomian decomposition method. Math. Chem. 52(1), 255–267 (2014)R. Singh, G. Nelakanti, J. Kumar, A new effcient technique for solving two-point boundary value problems for integro-differential equations. Math. Chem. 52, 2030–2051 (2014

On improved three-step schemes with high efficiency index and their dynamics

This paper presents an improvement of the sixth-order method of Chun and Neta as a class of three-step iterations with optimal efficiency index, in the sense of Kung-Traub conjecture. Each member of the presented class reaches the highest possible order using four functional evaluations. Error analysis will be studied and numerical examples are also made to support the theoretical results. We then present results which describe the dynamics of the presented optimal methods for complex polynomials. The basins of attraction of the existing optimal methods and our methods are presented and compared to illustrate their performances.This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02 and FONDOCYT Republica Dominicana.Babajee, DKR.; Cordero Barbero, A.; Soleymani, F.; Torregrosa Sánchez, JR. (2014). On improved three-step schemes with high efficiency index and their dynamics. Numerical Algorithms. 65(1):153-169. https://doi.org/10.1007/s11075-013-9699-6S153169651Pang, J.S., Chan, D.: Iterative methods for variational and complementary problems. Math. Program. 24(1), 284–313 (1982)Sun, D.: A class of iterative methods for solving nonlinear projection equations. J. Optim. Theory Appl. 91(1), 123–140 (1996)Chun, C., Neta, B.: A new sixth-order scheme for nonlinear equations. Appl. Math. Lett. 25, 185–189 (2012)Kung, H.T., Traub, J.F.: Optimal order of one-point and multipoint iteration. J. ACM 21, 643–651 (1974)Neta, B.: A new family of high-order methods for solving equations. Int. J. Comput. Math. 14, 191–195 (1983)Neta, B.: On Popovski’s method for nonlinear equations. Appl. Math. Comput. 201, 710–715 (2008)Chun, C., Neta, B.: Some modifications of Newton’s method by the method of undeterminate coefficients. Comput. Math. Appl. 56, 2528–2538 (2008)Chun, C., Lee, M.Y., Neta, B., Dzunic, J.: On optimal fourth-order iterative methods free from second derivative and their dynamics. Appl. Math. Comput. 218, 6427–6438 (2012)Cordero, A., Torregrosa, J.R., Vassileva, M.P.: Three-step iterative methods with optimal eighth-order convergence. J. Comput. Appl. Math. 235, 3189–3194 (2011)Cordero, A., Torregrosa, J.R., Vassileva, M.P.: A family of modified Ostrowski’s methods with optimal eighth order of convergence. Appl. Math. Lett. 24, 2082–2086 (2011)Heydari, M., Hosseini, S.M., Loghmani, G.B.: On two new families of iterative methods for solving nonlinear equations with optimal order. Appl. Anal. Dis. Math. 5, 93–109 (2011)Neta, B., Petkovic, M.S.: Construction of optimal order nonlinear solvers using inverse interpolation. Appl. Math. Comput. 217, 2448–2445 (2010)Sharifi, M., Babajee, D.K.R., Soleymani, F.: Finding the solution of nonlinear equations by a class of optimal methods. Comput. Math. Appl. 63, 764–774 (2012)Soleymani, F., Karimi Vanani, S., Khan, M., Sharifi, M.: Some modifications of King’s family with optimal eighth order of convergence. Math. Comput. Model. 55, 1373–1380 (2012)Soleymani, F., Karimi Vanani, S., Jamali Paghaleh, M.: A class of three-step derivative-free root solvers with optimal convergence order. J. Appl. Math. 2012, Article ID 568740, 15 pp. (2012). doi: 10.1155/2012/568740Soleymani, F., Sharifi, M., Mousavi, B.S.: An improvement of Ostrowski’s and King’s techniques with optimal convergence order eight. J. Optim. Theory Appl. 153, 225–236 (2012)Stewart, B.D.: Attractor basins of various root-finding methods. M.S. Thesis, Naval Postgraduate School, Department of Applied Mathematics, Monterey, CA (2001)Amat, S., Busquier, S., Plaza, S.: Review of some iterative root-finding methods from a dynamical point of view. Scientia 10, 3–35 (2004)Amat, S., Busquier, S., Plaza, S.: Dynamics of the King and Jarratt iterations. Aequ. Math. 69, 212–223 (2005)Amat, S., Busquier, S., Plaza, S.: Chaotic dynamics of a third-order Newton type method. J. Math. Anal. Appl. 366, 24–32 (2010)Neta, B., Chun, C., Scott, M.: A note on the modified super-Halley method. Appl. Math. Comput. 218, 9575–9577 (2012)Scott, M., Neta, B., Chun, C.: Basin attractors for various methods. Appl. Math. Comput. 218, 2584–2599 (2011)Ardelean, G.: A comparison between iterative methods by using the basins of attraction. Appl. Math. Comput. 218, 88–95 (2011)Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice Hall, New York (1964)Babajee, D.K.R.: Analysis of higher order variants of Newton’s method and their applications to differential and integral equations and in ocean acidification. Ph.D. Thesis, University of Mauritius (2010